Cauchy - Riemann Function in terms of Z

In summary, the conversation is about finding the conjugate harmonic function for U(x,y)=sin(x)cosh(y) and simplifying it in terms of Z. The individual is attaching their work and asking for guidance on the simplification process. They have used trigonometric substitutions but there may still be a sign problem present.
  • #1
KleZMeR
127
1

Homework Statement



I found the function V, which is the conjugate harmonic function for U(x,y)=sin(x)cosh(y). I am attaching my work. It turns out to be a two-term function with trig factors. I am then to write F(Z) in terms of Z, but is plugging in x, and y, in terms of Z into my trig functions good enough? I think there's some simplification that can take place, i.e. Euler, as I started, but I am just wondering if there is a specific direction I should take to simplify before I crunch the math? There are many directions and this is my first problem like this. Any help would be appreciated.


Homework Equations





The Attempt at a Solution

 

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  • #2
Ok, I think I simplified it in terms of Z, if anyone disagrees please let me know! I used cosh(y)=cos(i*y), and i*sinh(x)=sin(i*x) , and another often-used trig sub.
 

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  • #3
KleZMeR said:
Ok, I think I simplified it in terms of Z, if anyone disagrees please let me know! I used cosh(y)=cos(i*y), and i*sinh(x)=sin(i*x) , and another often-used trig sub.

The harmonic conjugate of a function U is supposed the be the imaginary part of an analytic function where U is the real part. sin(z*) is NOT analytic. You have a sign problem and from your posted photos I can't tell where it came from. Try and show your steps in TeX. Or find it yourself.
 
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  • #4
Thanks Dick, yes it was a sign problem.
 

FAQ: Cauchy - Riemann Function in terms of Z

What is the Cauchy-Riemann function in terms of Z?

The Cauchy-Riemann function in terms of Z is a complex-valued function that satisfies the Cauchy-Riemann equations, which are a set of partial differential equations that describe the analyticity of a function in the complex plane. In other words, the function is differentiable at every point in the complex plane and has a complex derivative.

How is the Cauchy-Riemann function related to complex differentiability?

The Cauchy-Riemann function is closely related to complex differentiability. If a function satisfies the Cauchy-Riemann equations, it is said to be analytic or holomorphic, which means it is infinitely differentiable at every point in the complex plane. Conversely, if a function is analytic, it automatically satisfies the Cauchy-Riemann equations.

How do the Cauchy-Riemann equations relate to geometric properties?

The Cauchy-Riemann equations have important geometric implications. They state that the real and imaginary parts of an analytic function must satisfy the Laplace equation, which is a special case of the more general Cauchy-Riemann equations. This means that the function's level curves (contours of constant real and imaginary parts) must be orthogonal at every point in the complex plane.

Can the Cauchy-Riemann function be used to solve problems in physics or engineering?

Yes, the Cauchy-Riemann function has many applications in physics and engineering. For example, it is used in fluid dynamics to describe the flow of a fluid in the complex plane. It is also used in electrical engineering to solve problems involving AC circuits and signal processing. In addition, the Cauchy-Riemann equations have been applied to problems in optics, elasticity, and other areas of physics and engineering.

What is the significance of the Cauchy-Riemann function in complex analysis?

The Cauchy-Riemann function is of utmost importance in complex analysis. It plays a key role in the development of the theory of complex functions, including the concept of analytic functions, Cauchy's integral theorem and formula, and the Cauchy integral formula. The Cauchy-Riemann equations also have many applications in the study of conformal mappings and the behavior of complex functions near singularities.

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